Universal Critical Behavior of Aperiodic Ferromagnetic Models

TL;DR

This paper explores how geometric fluctuations in aperiodic ferromagnetic models affect their critical behavior, revealing new universality classes and critical exponents through exact recursion relations and numerical analysis.

Contribution

It introduces a detailed analysis of aperiodic exchange interactions on hierarchical lattices, identifying new critical behaviors and calculating associated critical exponents.

Findings

Existence of a non-trivial fixed point for irrelevant fluctuations.

Emergence of a two-cycle and a new critical behavior for relevant fluctuations.

Critical exponent α differs from the uniform case and matches numerical results.

Abstract

We investigate the effects of geometric fluctuations, associated with aperiodic exchange interactions, on the critical behavior of $q$-state ferromagnetic Potts models on generalized diamond hierarchical lattices. For layered exchange interactions according to some two-letter substitutional sequences, and irrelevant geometric fluctuations, the exact recursion relations in parameter space display a non-trivial diagonal fixed point that governs the universal critical behavior. For relevant fluctuations, this fixed point becomes fully unstable, and we show the apperance of a two-cycle which is associated with a novel critical behavior. We use scaling arguments to calculate the critical exponent $\alpha$ of the specific heat, which turns out to be different from the value for the uniform case. We check the scaling predictions by a direct numerical analysis of the singularity of the thermodynamic free-energy. The agreement between scaling and direct calculations is excellent for stronger singularities (large values of $q$). The critical exponents do not depend on the strengths of the exchange interactions.